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Abby's investment at 14% earned the same interest as Steve's which was $1500 less but at 18%. How much did steve invest?

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3 Answers

You need to use the formula for compound interest, A=P(1 + r/n)nt where r = annual interest rate, n= number of payments per year, and t= number of years. Some textbooks define n and t differently or use different variables. The interest earned will be the total amount - principal, or A - P.

To simplify this problem we will assume that interest is paid once per year, and that this is just for one year. So n=1 and t=1. Our formula becomes  A=P(1+r). So to calculate interest earned, A - P = P(1+r) - P = r(P)

For Abby, r = 0.14 and for Steve r= 0.18.

Let P = amount Abby invested. (P-1500) = Steve's investment

For Abby, the interest she earned = 0.14P

For Steve, the interest he earned =0.18(P-1500) = 0.18P - 220

Abby's interest = Steve's interest so,

0.14P = 0.18P -220 

Solve for P.

 -0.04P = -220

P = 5500   Are we done? No. P= Abby's investment and we need Steve's investment

Steve's investment = P - 1500 = $4000

Let 'x' represent the amount Abby invested at 14% and let 'y' represent the amount Steve invested at 18%. That is,

          amount Abby invested = x   

         amount Steve invested = y

Since the problem states that the amount Steve invested (y) is $1500 less than the amount Abby invested (x), then 

         y = x - 1500

Given that Abby's investment was at a 14% interest rate and Steve's investment was at an 18% interest rate, then

        interest Abby earned from her investment = (14%/100%)·x = 0.14x

        interest Steve earned from his investment = (18%/100%)·y = 0.18y

Since their investments earned the same interest at their respective rates, then we can set the equation of the interest they earned from their investment equal to one another:

          0.14x = 0.18y

Since there are 2 unknown variables, x and y, we need to solve for one variable in order to solve for the other. To do so, given that   y = x - 1500 ,  we can solve for x by substituting this expression for y in the equation above:

          0.14x = 0.18(x - 1500)

Expanding the right hand side of the equation we arrive at the following:

          0.14x = 0.18x - 270

Subtract 0.18x from both sides of the equation:

          0.14x - 0.18x = 0.18x - 0.18x - 270

                    -0.04x = -270

Divide both sides by -0.04 to solve for x:

          -0.04x/-0.04 = -270/-0.04

                          x = 6750

Abby invested $6750, but we are looking to find how much Steve invested (y). So we solve for y by substituting the value determined for x into the equation for y:

          y = x - 1500

          y = 6750 - 1500

          y = 5250

Thus, Steve invested $5250.